Pythagoras has been attributed with the development of a musical scale based upon harmony or consonance between two musical notes (interval) when the higher note relative to the lower note is an eighth (octave--frequency ratio 2:1), a fifth (frequency ratio 3:2) or a fourth (frequency ratio 4:3). Lesser consonant sixth (frequency ratio 5:3) and third (frequency ratio 5:4) and generally dissonant second (frequency ratio 9:8) and seventh (frequency ratio 15:8) complete the scale known as the seven note diatonic scale labeled with the seven letters A, B, C, D, E, F and G. Each eighth note or octave begins a new series. The diatonic scale evolved into the twelve note chromatic scale commonly used today wherein semitones or half steps are added between the notes with larger intervals, i.e., A#, C#, D#, F# and G#, to form a scale entirely of half steps.
Each note or music tone contains a fundamental frequency (first partial) together with a plurality of harmonic frequencies (second, third, etc., partials). The consonance or dissonance (harmony or harshness) between two simultaneous notes (interval) generally depends upon the interactions of the partials of one note with the other note. Dissonance occurs when there is a small frequency difference between two partials. Differences in the range from about three to ten hertz produce audible beats while somewhat larger frequency differences, depending upon the frequency, also produce harshness. For an octave interval, all the partials of the higher note coincide with partials of the lower note producing a high degree of consonance. For a fifth interval, the second, fourth, sixth, etc. partials of the higher note coincide with the third, fifth, seventh, etc. partials of the lower note while the differences between the other partials are sufficient to avoid dissonance thus producing consonance. For a fourth interval, the third, sixth, ninth, etc. partials of the higher note coincide with the fourth, seventh, tenth, etc. partials of the lower note while the differences between the other partials are sufficient to avoid dissonance thus producing consonance. Eighths, fifths and fourths are called perfect intervals because of their consonance.
Generally the fundamental frequencies F(n) in the twelve note chromatic scale are in accordance with the equation EQU F.sub.n =F.sub.0 *2.sup.n/12
wherein F.sub.0 is the first frequency of the scale and n is the number of the note from F.sub.0. It is noted that this equation does not give the exact relationship 3:2 for a fifth (n=7) or 4:3 for a fourth (n=5) but is sufficiently close to be considered consonant and is herein considered the same.
Musical instruments having keyboards for producing notes normally include a linear array of white keys arranged in the diatonic scale with raised black keys disposed between the rear portions of selected white keys for adding the half steps to complete the twelve note chromatic scale. One type of keyboard instrument has an electronic note generator which responds to a depressed key to produce the corresponding musical note.
String instruments, such as a guitar, violin, cello, string bass, etc., have a plurality of parallel strings with a plurality of frets extending perpendicular to the strings so that each string can be held against a fret to increase the vibration frequency of each string. The spacing of the frets is generally designed to produce half-step changes in string frequency in accordance with the chromatic scale when the string is pressed against successive frets (stopped). The masses and tensions of the different strings are selected to produce intervals such as fourths, fifths, etc. between strings. The spacing of frets on conventional string instruments with one-half step produces corresponding stopped string lengths L(n) of ##EQU1## wherein L.sub.0 is the length of the unstopped string and n is the number of the fret from the distal end of the finger board.
Another type of string instrument known as the hammer dulcimer has a plurality of parallel strings of different lengths defining half steps. Engageable left and right bridges define left and right string portions for producing intervals equal to one or two octaves.
Various scales other than the twelve note chromatic scale of half notes have been used in musical instruments. Among these are seventeen, twenty-two and twenty-four note scales. The twenty-four note scale consists of quarter steps and, if adopted, would provide musicians with many additional musical effects from intervals, chords, and other multiple note combinations. Pentatonic scales have been employed in music and include five notes per octave based upon fourth intervals.